Higher-order derivatives of generalized Lyapunov-like functions for switched nonlinear systems

This paper is concerned with utilizing higher-order derivatives of generalized Lyapunov-like functions to infer asymptotic stability of the switched nonlinear system without requiring the first derivative of the Lyapunov function for each activated subsystem to be negative definite. In contrast to the existing switched systems literature, we allow generalized Lyapunov-like functions to increase both during the continuous flows and the discrete jumps. Firstly, a set of sufficient conditions for asymptotic stability of switched nonlinear systems are derived under a state-dependent switching law. Then, all the above-mentioned conditions are transformed to a set of bilinearly diagonally dominant sum of squares constraints. Bilinearly diagonally dominant sum of squares programming is proposed as a tool to solve the conditions provided. Finally, two examples are introduced to prove the effectiveness of the above method.

Automated summary

This paper proposes a new method to infer asymptotic stability of switched nonlinear systems using higher-order derivatives of generalized Lyapunov-like functions. By introducing bilinearly diagonally dominant sum of squares constraints, the stability conditions are transformed into a more computationally tractable form.